Consider the earth orbiting around the sun. We know the
interaction and can solve for the orbit.

From
the force you calculate the earth’s orbit and with initial condition you know
where it is at all times.

Given the Hamitonian H one can
calculate the quantum wave function that is equivalent to the full QS.

Know how the atom will behave in terms of the wave function.

Suppose the problem is too difficult to solve because the
interaction is more complicated.

For the two slit experiment you propagate a particle to slit
1 (the interaction) and then to the screen.

For this I can write down an amplitude
to reach the screen by nature of the interaction.

One difference in QM is that I need to find all possible
ways to reach the ending state

I can imagine a more complex system with a series of
interactions.

My goal would be to calculate the amplitude to travel along
any given path and then sum over all possible paths.Feynman diagrams diagrammatically represent this
deeply complicated mathematical process. A first order diagram in some sense shows the
interaction as a single step process, as if you can get a good result by
correcting the trajectory one time classically. High order diagrams are similar
to the multiple corrections required to find the asteroids final trajectory.In addition to the expansion of the interaction
Feynman diagrams represent the multitude of ways that quantum systems can travel.
The amplitudes for each trajectory will be included and interference may
result.